The integration of polynomial Liénard systems by elementary functions (Q1387152)

A polynomial Liénard system is a pair of simultaneous ordinary differential equations \[ \dot{x}=y, \qquad\dot{y}=-f(x)y-g(x) \] for \(f,g\in\mathbb{C}[x]\) of degree \(m\) and \(n\) respectively. The main result here is that for \(f,g\) nonzero and \(n\leq m\) the system has a nonconstant elementary first integral if and only if \(g\) is a constant multiple of \(f\). For the range \(m<n\leq 2m\) a similar, more technical result is proved. For \(n>2m+1\) it is shown that no such integral exists. The method uses known results of \textit{M. J. Prelle} and \textit{M. F. Singer} [Trans. Am. Math. Soc. 279, 215-229 (1983; Zbl 0527.12016)] and from a recent unpublished work of Zołądek. An application of the results to Bogdanov-Takens system is described, as is Wilson's example for the case \(n=2m+1\). \textit{M. Hayashi} [Funkc. Ekvacioj, Ser. Int. 39, No. 3, 403-408 (1996; Zbl 0871.34030)] has done related work in the case \(m+1\geq n\).